Computing device and method for analyzing scattering parameters passivity

ABSTRACT

A computing device and a method measures scattering parameters (S-parameters) values at ports of a circuit at different signal frequencies, and creates a non-common-pole rational function of S-parameters by applying a vector fitting algorithm to the S-parameters. A matrix of the non-common-pole rational function is converted to a state-space matrix, and the state-space matrix is substituted into a Hamiltonian matrix. The device and method further analyzes if eigenvalues of the Hamiltonian matrix have pure imaginaries, to determine if the non-common-pole rational function of the S-parameters satisfies a passivity requirement.

BACKGROUND

1. Technical Field

Embodiments of the present disclosure relates to circuit simulatingmethods, and more particularly, to a computing device and a method foranalyzing scattering parameters (S-parameters) passivity.

2. Description of Related Art

Scattering parameters (S-parameters) are useful for analyzing behaviourof circuits without regard to detailed components of the circuits. TheS-parameters may be measured at ports of a circuit at different signalfrequencies. In a high frequency and microwave circuit design, theS-parameters of the circuit may be used to create a rational function,and the rational function may be used to generate an equivalent circuitmodel, which may be applied to time-domain analysis for the circuitdesign. For judging whether the circuit design satisfies stabilityrequirements, the time-domain analysis result should be convergent. Toensure constringency, the rational function and the equivalent circuitmodel of the S-parameters are required to be passive.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of one embodiment of a computing device foranalyzing S-parameters passivity.

FIG. 2 is a block diagram of one embodiment of function modules of ananalysis unit in the computing device of FIG. 1.

FIG. 3 is a flowchart of one embodiment of a method for analyzingS-parameters passivity.

FIG. 4 is one embodiment of four-port differential transmission lines ina circuit.

FIG. 5 is one embodiment of a common-pole rational function and anon-common-pole rational function of the S-parameters measured at theports shown in FIG. 4.

FIG. 6 is one embodiment of pure imaginaries of a Hamiltonian matrix.

DETAILED DESCRIPTION

The disclosure, including the accompanying drawings in which likereferences indicate similar elements, is illustrated by way of examplesand not by way of limitation. It should be noted that references to “an”or “one” embodiment in this disclosure are not necessarily to the sameembodiment, and such references mean at least one.

In general, the word “module,” as used hereinafter, refers to logicembodied in hardware or firmware, or to a collection of softwareinstructions, written in a programming language, such as, for example,Java, C, or Assembly. One or more software instructions in the modulesmay be embedded in firmware. It will be appreciated that modules maycomprised connected logic units, such as gates and flip-flops, and maycomprise programmable units, such as programmable gate arrays orprocessors. The modules described herein may be implemented as eithersoftware and/or hardware modules and may be stored in any type ofcomputer-readable medium or other computer storage device.

FIG. 1 is a block diagram of one embodiment of a computing device 30.The computing device 30 is connected to a measurement device 20. Themeasurement device 20 measures S-parameters at ports of a circuit 10, toobtain an S-parameters file 32, and stores the S-parameters file 32 in astorage device 34 of the computing device 30. Depending on theembodiment, the storage device 34 may be a smart media card, a securedigital card, or a compact flash card. The measurement device 20 may bea network analyzer. The computing device 30 may be a personal computer,or a server, for example.

In this embodiment, the computing device 30 further includes an analysisunit 31 and a processor 33. The analysis unit 31 includes a number offunction modules (depicted in FIG. 2) The function modules may comprisecomputerized code in the form of one or more programs that are stored inthe storage device 34. The computerized code includes instructions thatare executed by the at least one processor 33, to create anon-common-pole rational function of the S-parameters, generate anequivalent circuit model 315 of the circuit 10 according to thenon-common-pole rational function, and analyze if non-common-polerational function satisfies passivity requirements, so as to determineif the equivalent circuit model 315 satisfies passivity requirements.

FIG. 2 is a block diagram of the function modules of the analysis unit31 in the computing device of FIG. 1. The analysis unit 31 includes aparameter reading module 311, a vector fitting module 312, a matrixconversion module 313, and a passivity analysis module 314. Adescription of functions of the modules 311 to 314 are included in thefollowing description of FIG. 3.

FIG. 3 is a flowchart of one embodiment of a method for analyzingS-parameters passivity. Depending on the embodiment, additional blocksmay be added, others removed, and the ordering of the blocks may bechanged.

In block S301, the measurement device 20 measures the S-parameters atthe ports of the circuit 10, to obtain the S-parameters file 32, andstores the S-parameters file 32 in the storage device 34. The parameterreading module 311 reads the S-parameters file 32 from the storagedevice 34. In one embodiment, the S-parameters include reflectioncoefficients, insertion loss coefficients, near-end crosstalkcoefficients, and remote-end crosstalk coefficients of each port in thecircuit 10. The S-parameters file 32 records S-parameter values measuredat ports of the circuit 10 at different signal frequencies. For example,suppose a differential transmission line 11 of the circuit 10 includesfour ports numbered 1, 2, 3, and 4 as shown in FIG. 4. When a testsignal is input to the port 1, the port 1 will receive a reflectionsignal, and the ports 2, 3, 4 will respectively receive a first, asecond, and a third transmission signal. Then, a ratio of the reflectionsignal power and the test signal power is regarded as a reflectioncoefficient S11, a ratio of the second transmission signal power and thetest signal power is regarded as an insertion loss coefficient S12, aratio of the third transmission signal power and the test signal poweris regarded as a near-end crosstalk coefficient S13, and a ratio of thethird transmission signal power and the test signal power is regarded asa remote-end crosstalk coefficient S14. When a frequency of the testsignal changes, values of the ratios change, so the S-parameters file 32records a large quantity of S-parameter values.

In block S303, the vector fitting module 312 creates a non-common-polerational function of the S-parameters by applying a vector fittingalgorithm to the S-parameters, and generates the equivalent circuitmodel 315 of the circuit 10 according to the non-common-pole rationalfunction.

It is understood that, two modes of rational functions may be createdfor the S-parameters by applying the vector fitting algorithm to theS-parameters. One is the common-pole style labeled (1a), the other isthe non-common pole style labeled (1b).

$\begin{matrix}{{S(s)} \approx {\left( {\sum\limits_{m = 1}^{M}\frac{\begin{bmatrix}r_{m}^{1,1} & r_{m}^{1,2} & \ldots & r_{m}^{1,N} \\r_{m}^{2,1} & r_{m}^{2,2} & \ldots & r_{m}^{2,N} \\\ldots & \ldots & \ldots & \ldots \\r_{m\;}^{N,1} & r_{m}^{N,2} & \ldots & r_{m}^{N,N}\end{bmatrix}}{s + p_{m}}} \right) + \begin{bmatrix}d_{m}^{1,1} & d_{m}^{1,2} & \ldots & d_{m}^{1,N} \\d_{m\;}^{2,1} & d_{m}^{2,2} & \ldots & d_{m}^{2,N} \\\ldots & \ldots & \ldots & \ldots \\d_{m}^{N,1} & d_{m}^{N,2} & \ldots & d_{m}^{N,N}\end{bmatrix}}} & \left( {1a} \right) \\{{S(s)} \approx {\left( {\sum\limits_{m = 1}^{M}\begin{bmatrix}\frac{r_{m}^{1,1}}{s + p_{m}^{1,1}} & \frac{r_{m}^{1,2}}{s + p_{m}^{1,2}} & \ldots & \frac{r_{m}^{1,N}}{s + p_{m}^{1,N}} \\\frac{r_{m}^{2,1}}{s + p_{m}^{2,1}} & \frac{r_{m}^{2,2}}{s + p_{m}^{2,2}} & \ldots & \frac{r_{m}^{2,N}}{s + p_{m}^{2,N}} \\\ldots & \ldots & \ldots & \ldots \\\frac{r_{m}^{N,1}}{s + p_{m}^{N,1}} & \frac{r_{m}^{N,2}}{s + p_{m}^{N,2}} & \ldots & \frac{r_{m}^{N,N}}{s + p_{m}^{N,N}}\end{bmatrix}} \right) + {\quad\begin{bmatrix}d_{m}^{1,1} & d_{m}^{1,2} & \ldots & d_{m}^{1,N} \\d_{m}^{2,1} & d_{m}^{2,2} & \ldots & d_{m}^{2,N} \\\ldots & \ldots & \ldots & \ldots \\d_{m}^{N,1} & d_{m}^{N,2} & \ldots & d_{m}^{N,N}\end{bmatrix}}}} & \left( {1b} \right)\end{matrix}$

where M represents a control precision of the function (eithercommon-pole or non-common-pole), N represents a number of the ports ofthe circuit 10, r_(m) represents residue values, p_(m) represents polevalues, s=ω=2 πf, f represents a frequency of the test signal, and d_(m)represents a constant. It is understood that the control precision meanshow many pairs of pole-residue values are utilized in the function (1a)or (1b).

FIG. 5 shows a real line and two broken lines. The real line representsan original curve of the S-parameters. The two broken lines representcurves of the rational functions (1a) and (1b), which utilize eightpairs of pole-residue values in the vector fitting algorithm of theS-parameters of the four ports shown in FIG. 4. In FIG. 5, thehorizontal axis represents the frequency of the test signal, thevertical axis represents an amplitude of the test signal. It can be seenin FIG. 5 that the curve of the non-common-pole function approaches theoriginal curve much better than the curve of the common-pole function,which means the non-common-pole function has a higher fitting precisionthan the common-pole function. Therefore, in this embodiment, thenon-common-pole rational function is used for analyzing passivity of theS-parameters.

A matrix of the non-common-pole rational function labeled as (1b) is asfollows:

$\begin{matrix}{{{S(s)} \approx {\hat{S}(s)}} = \begin{bmatrix}{{\hat{S}}_{11}(s)} & {{\hat{S}}_{12}(s)} & \ldots & {{\hat{S}}_{1N}(s)} \\{{\hat{S}}_{21}(s)} & {{\hat{S}}_{22}(s)} & \ldots & {{\hat{S}}_{2N}(s)} \\\ldots & \ldots & \ldots & \ldots \\{{\hat{S}}_{N\; 1}(s)} & {{\hat{S}}_{N\; 2}(s)} & \ldots & {{\hat{S}}_{NN}(s)}\end{bmatrix}} & (2)\end{matrix}$

where Ŝ_(p,q)(s)=Ŝr_(p,q)(s)+Ŝc_(p,q)(s)+d^(p,q), Ŝr_(p,q)(s) representspole values, and a real number expression function of Ŝr_(p,q)(s) islabeled (3a). Ŝc_(p,q)(s) represents residue values, and a complexnumber expression function of Ŝc_(p,q)(s) is labeled (3b).

$\begin{matrix}{{\hat{S}{r_{p,q}(s)}} = {\sum\limits_{u = 1}^{U}\frac{r_{u}^{p,a}}{s + p_{u}^{p,q}}}} & \left( {3a} \right)\end{matrix}$

where p_(u) ^(p,q) and r_(u) ^(p,q) are real numbers.

$\begin{matrix}{{\hat{S}{c_{p,q}(s)}} = {{\sum\limits_{v = 1}^{V}\frac{{{Re}\left( r_{v}^{p,q} \right)} + {{{Im}\left( r_{v}^{p,q} \right)}j}}{s + {{Re}\left( p_{v}^{p,q} \right)} + {{{Im}\left( p_{v}^{p,q} \right)}j}}} + \frac{{{Re}\left( r_{v}^{p,q} \right)} - {{{Im}\left( r_{v}^{p,q} \right)}j}}{s + {{Re}\left( p_{v}^{p,q} \right)} - {{{Im}\left( p_{v}^{p,q} \right)}j}}}} & \left( {3b} \right)\end{matrix}$

where j=√{square root over (−1)}, U+2V=M, p_(u) ^(p,q)>0, and Re(p_(v)^(p,q))>0.

In block S305, the matrix conversion module 313 converts the matrix ofthe non-common-pole rational function to a state-space matrix. Forexample, the matrix conversion module 313 combines the matrix (2) andthe expression functions (3a) and (3b) to obtain a combined matrix, thenconverts the combined matrix into the state-space matrix as follows:jωX(jω)=AX(jω)+BU(jω)Y(jω)=CX(jω)+DU(jω)  (4)

Where the matrixes A, B, C, and D may be expressed as follows:

$\begin{matrix}{{A = \begin{bmatrix}A_{r} & 0 \\0 & A_{c}\end{bmatrix}},{B = \begin{bmatrix}B_{r} \\B_{c}\end{bmatrix}},{C = \begin{bmatrix}C_{r} & C_{c}\end{bmatrix}},{D = \begin{bmatrix}d^{1,1} & d^{1,2} & \ldots & d^{1,N} \\d^{2,1} & d^{2,2} & \ldots & d^{2,N} \\\ldots & \ldots & \ldots & \ldots \\d^{N,1} & d^{N,1} & \ldots & d^{N,N}\end{bmatrix}}} & (5)\end{matrix}$

where A_(r), B_(r) and C_(r) are state-space matrixes of the pole valuesŜr_(p,q)(s) in (3a), and may be expressed as follows:

$\begin{matrix}{{A_{r}\left( {i,j} \right)} = \left\{ {{\begin{matrix}{p_{0}^{p,q},} & {i = {{j\mspace{14mu}{and}\mspace{14mu} i} = {{\left( {U \cdot N} \right)\left( {p - 1} \right)} + {U\left( {q - 1} \right)} + u}}} \\0 & {o.w.}\end{matrix}{B_{r}\left( {i,j} \right)}} = \left\{ {{\begin{matrix}{1,} & {i = {{{\left( {U \cdot N} \right)\left( {p - 1} \right)} + {U\left( {q - 1} \right)} + {u\mspace{14mu}{and}\mspace{14mu} j}} = q}} \\0 & {o.w.}\end{matrix}{C_{r}\left( {i,j} \right)}} = \left\{ \begin{matrix}{r_{u}^{p,q},} & {i = {{p\mspace{14mu}{and}\mspace{14mu} j} = {{\left( {U \cdot N} \right)\left( {p - 1} \right)} + {U\left( {q - 1} \right)} + u}}} \\0 & {o.w.}\end{matrix} \right.} \right.} \right.} & (6)\end{matrix}$

where A_(r) is a (N·N·U)×(N·N·U) sparse matrix, B_(r) is a (N·U)×Nsparse matrix, C_(r) is a N×(N·U) sparse matrix.

A_(c), B_(c), and C_(c) are state-space matrixes of the residue valuesŜc_(p,q)(s) in (3b), and may be expressed as follows:

$\begin{matrix}{{A_{c}\left( {i,j} \right)} = \left\{ {{\begin{matrix}{{{Re}\left( p_{v}^{p,q} \right)},} & {{i = {{j\mspace{14mu}{and}\mspace{14mu} i} = {\Psi\left( {p,q,v} \right)}}},{i = {{\Psi\left( {p,q,v} \right)} - 1}}} \\{{{Im}\left( p_{v}^{p,q} \right)},} & {{i = {{\Psi\left( {p,q,v} \right)} - 1}},{j = {\Psi\left( {p,q,v} \right)}}} \\{{- {{Im}\left( p_{v}^{p,q} \right)}},} & {{i = {\Psi\left( {p,q,v} \right)}},{j = {{\Psi\left( {p,q,v} \right)} - 1}}} \\{0,} & {o.w.}\end{matrix}{B_{c}\left( {i,j} \right)}} = \left\{ {{\begin{matrix}{2,} & {i = {{{\Psi\left( {p,q,v} \right)} - {1\mspace{14mu}{and}\mspace{14mu} j}} = q}} \\0 & {o.w.}\end{matrix}{C_{c}\left( {i,j} \right)}} = \left\{ \begin{matrix}{{{Re}\left( r_{v}^{p,q} \right)},} & {i = {{p\mspace{14mu}{and}\mspace{14mu} j} = {\Psi - 1}}} \\{{{Im}\left( r_{v}^{p,q} \right)},} & {i = {{p\mspace{14mu}{and}\mspace{14mu} j} = \Psi}} \\{0,} & {o.w.}\end{matrix} \right.} \right.} \right.} & (7)\end{matrix}$

where Ψ(p,q,v)=(2V·N)(p−1)+2V (q−1)+2v, A_(c) is a sparse matrix, B_(r)is a (N·2V)×N sparse matrix, and C_(r) is a N×(N·2V) sparse matrix.

The matrix conversion module 313 combines the functions (5), (6), and(7) to obtain expressions of the coefficients A, B, C, and D in thestate-space matrix (4).

In block S307, the passivity analysis module 314 substitutes thestate-space matrix (4) into a Hamiltonian matrix, where the Hamiltonianmatrix is as follows:

$\begin{matrix}{H = \begin{bmatrix}{A - {{BR}^{- 1}D^{T}C}} & {{- {BR}^{- 1}}B^{T}} \\{C^{T}Q^{- 1}C} & {{- A^{T}} + {C^{T}{DR}^{- 1}B^{T}}}\end{bmatrix}} & (8)\end{matrix}$

where R=D^(T)D−I Q=DD^(T)−I, I is an identity matrix.

In block S309, the passivity analysis module 314 analyzes theeigenvalues of the Hamiltonian matrix for pure imaginaries, to determineif the non-common-pole rational function of the S-parameters satisfies apassivity requirement. If the eigenvalues of the Hamiltonian matrix havepure imaginaries, the procedure goes to block S311, the passivityanalysis module 314 determines that the non-common-pole rationalfunction of the S-parameters satisfies the passivity requirement. Forexample, eigenvalues of the Hamiltonian matrix (8) have pure imaginariesshown in FIG. 6, therefore, the non-common-pole rational function (2) ofthe S-parameters is passive. Accordingly, the passivity analysis module314 determines that the equivalent circuit model 315 satisfies thepassivity requirement.

Otherwise, if the eigenvalues of the Hamiltonian matrix have no pureimaginaries, the procedure goes to block S313, the passivity analysismodule 314 determines that the non-common-pole rational function of theS-parameters does not satisfy the passivity requirement. Accordingly,the passivity analysis module 314 determines that the equivalent circuitmodel 315 does not satisfy the passivity requirement.

Although certain inventive embodiments of the present disclosure havebeen specifically described, the present disclosure is not to beconstrued as being limited thereto. Various changes or modifications maybe made to the present disclosure without departing from the scope andspirit of the present disclosure.

What is claimed is:
 1. A computing device, comprising: a storage device;at least one processor; and an analysis unit comprising one or morecomputerized codes, which are stored in the storage device andexecutable by the at least one processor, the one or more computerizedcodes comprising: a parameter reading module operable to read ascattering parameters (S-parameters) file from the storage device,wherein the S-parameters file records S-parameters values measured atports of a circuit at different signal frequencies; a vector fittingmodule operable to create a non-common-pole rational function of theS-parameters by applying a vector fitting algorithm to the S-parameters;a matrix conversion module operable to convert a matrix of thenon-common-pole rational function to a state-space matrix; a passivityanalysis module operable to substitute the state-space matrix into aHamiltonian matrix, analyze if eigenvalues of the Hamiltonian matrixhave pure imaginaries, determine that the non-common-pole rationalfunction of the S-parameters satisfies a passivity requirement if theeigenvalues of the Hamiltonian matrix have pure imaginaries, ordetermine that the non-common-pole rational function of the S-parametersdoes not satisfy the passivity requirement if the eigenvalues of theHamiltonian matrix have no pure imaginaries.
 2. The computing device asclaimed in claim 1, wherein the vector fitting module is furtheroperable to generate an equivalent circuit model of the circuitaccording to the non-common-pole rational function.
 3. The computingdevice as claimed in claim 2, wherein the passivity analysis module isfurther operable to determine that the equivalent circuit modelsatisfies the passivity requirement if the non-common-pole rationalfunction of the S-parameters satisfies the passivity requirement, ordetermine the equivalent circuit model does not satisfy the passivityrequirement if the non-common-pole rational function of the S-parametersdoes not satisfy the passivity requirement.
 4. The computing device asclaimed in claim 1, wherein the S-parameters comprise reflectioncoefficients, insertion loss coefficients, near-end crosstalkcoefficients, and remote-end crosstalk coefficients of each port in thecircuit.
 5. The computing device as claimed in claim 1, wherein thestorage device is selected from the group consisting of a smart mediacard, a secure digital card, and a compact flash card.
 6. Acomputer-based method for analyzing scattering parameters (S-parameters)passivity, the method comprising: reading an S-parameters file from astorage device, wherein the S-parameters file records S-parametersvalues measured at ports of a circuit at different signal frequencies;creating a non-common-pole rational function of the S-parameters byapplying a vector fitting algorithm to the S-parameters; converting amatrix of the non-common-pole rational function to a state-space matrix;substituting the state-space matrix into a Hamiltonian matrix, andanalyzing if eigenvalues of the Hamiltonian matrix have pureimaginaries; and determining that the non-common-pole rational functionof the S-parameters satisfies a passivity requirement if the eigenvaluesof the Hamiltonian matrix have pure imaginaries, or determining that thenon-common-pole rational function of the S-parameters does not satisfythe passivity requirement if the eigenvalues of the Hamiltonian matrixhave no pure imaginaries.
 7. The method as claimed in claim 6, furthercomprising: generating an equivalent circuit model of the circuitaccording to the non-common-pole rational function.
 8. The method asclaimed in claim 7, further comprising: determining that the equivalentcircuit model satisfies the passivity requirement if the non-common-polerational function of the S-parameters satisfies the passivityrequirement, or determining that the equivalent circuit model does notsatisfy the passivity requirement if the non-common-pole rationalfunction of the S-parameters does not satisfy the passivity requirement.9. The method as claimed in claim 6, wherein the storage device isselected from the group consisting of a smart media card, a securedigital card, and a compact flash card.
 10. The method as claimed inclaim 6, wherein the S-parameters comprise reflection coefficients,insertion loss coefficients, near-end crosstalk coefficients, andremote-end crosstalk coefficients of each port of the differentialtransmission line.
 11. A non-transitory computer readable medium storinga set of instructions, the set of instructions capable of being executedby a processor of a computing device to perform a method for analyzingscattering parameters (S-parameters) passivity, the method comprising:reading a S-parameters file from the medium, wherein the S-parametersfile records S-parameters values measured at ports of a circuit atdifferent signal frequencies; creating a non-common-pole rationalfunction of the S-parameters by applying a vector fitting algorithm tothe S-parameters; converting a matrix of the non-common-pole rationalfunction to a state-space matrix; substituting the state-space matrixinto a Hamiltonian matrix, and analyzing if eigenvalues of theHamiltonian matrix have pure imaginaries; and determining that thenon-common-pole rational function of the S-parameters satisfies apassivity requirement if the eigenvalues of the Hamiltonian matrix havepure imaginaries, or determining that the non-common-pole rationalfunction of the S-parameters does not satisfy the passivity requirementif the eigenvalues of the Hamiltonian matrix have no pure imaginaries.12. The medium as claimed in claim 11, wherein the method furthercomprises: generating an equivalent circuit model of the circuitaccording to the non-common-pole rational function.
 13. The medium asclaimed in claim 12, wherein the method further comprises: determiningthat the equivalent circuit model satisfies the passivity requirement ifthe non-common-pole rational function of the S-parameters satisfies thepassivity requirement, or determining the equivalent circuit model doesnot satisfy the passivity requirement if the non-common-pole rationalfunction of the S-parameters does not satisfy the passivity requirement.14. The medium as claimed in claim 11, wherein the medium is selectedfrom the group consisting of a smart media card, a secure digital card,and a compact flash card.
 15. The medium as claimed in claim 11, whereinthe S-parameters comprise reflection coefficients, insertion losscoefficients, near-end crosstalk coefficients, and remote-end crosstalkcoefficients of each port of the differential transmission line.